Citation: Chavdarov, I.; Naydenov, B.
Algorithm for Determining the Types
of Inverse Kinematics Solutions for
Sequential Planar Robots and Their
Representation in the Configuration
Space. Algorithms 2022, 15, 469.
https://doi.org/10.3390/a15120469
Academic Editors: Shuai Li, Predrag
S. Stanimirovic, Dechao Chen,
Mohammed Aquil Mirza, Vasilios
N. Katsikis, Dunhui Xiao and
Frank Werner
Received: 19 September 2022
Accepted: 6 December 2022
Published: 9 December 2022
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Article
Algorithm for Determining the Types of Inverse Kinematics
Solutions for Sequential Planar Robots and Their
Representation in the Configuration Space
Ivan Chavdarov
1,2,
* and Bozhidar Naydenov
2,3
1
Faculty of Mathematics and Informatics, Department of Mechatronics, Robotics and Mechanics,
University of Sofia “St. Kliment Ohridski”, 1504 Sofia, Bulgaria
2
Institute of Robotics, Bulgarian Academy of Sciences, Acad. G. Bonchev St, Bl. 1, 1113 Sofia, Bulgaria
3
DASSAULT SYSTEMES, Blvd. General I. Totleben 53-55, 1606 Sofia, Bulgaria
* Correspondence: ivannc@uni-sofia.bg
Abstract:
The work defines in a new way the different types of solutions of the inverse kinematics
(IK) problem for planar robots with a serial topology and presents an algorithm for solving it. The
developed algorithm allows the finding of solutions for a wide range of robots by using a geometric
approach, representing points in a polar coordinate system. Inverse kinematics, which is one of the
most important, most studied and challenging problems in robotics, aims to calculate the values of the
joint variables, given the desired position and orientation of the robot’s end effector. Configuration
space is defined by joint angles and is the basis of most motion planning algorithms. Areas in the
working and configuration space are generated that are reachable with different types of solutions.
Programs are created that use the proposed algorithm for robots with two and three rotational
degrees of freedom, and graphically present the results in the workspace and configuration space.
The possibility of transitioning from one type of solution to another by passing through a singular
configuration is discussed. The results are important for planning motions in the workspace and
configuration space, as well as for the design and kinematic analysis of robots.
Keywords: inverse kinematics; robot; workspace and configuration space; singular configurations
1. Introduction
This paper presents an algorithm that examines the configuration spaces of planar
serial robots and uses this information to classify inverse kinematics solutions. Examples
are presented that show the benefit of dividing the workspace and configuration space into
zones with different types of solutions. The algorithm is implemented in AutoCAD and
can help visualize feasible trajectories. Advantages include finding solutions in singular
positions and automatically generating the configuration space along with the workspace.
Some limitations of the algorithm are that it is not applicable for real-time tasks and
currently can be used only for serial and planar robots.
1.1. Literature Review
Serial robots are the most common industrial manipulators. They are designed as a se-
ries of links connected by motor-actuated joints that extend from a base to an end-effector.
Robot kinematics deals with two types of problems—forward and inverse kinematics
problems. For serial robots the forward kinematics problem is straightforward, and a single
solution is easily found; the most popular method for the solution is proposed by Denavit
and Hartenberg [
1
]. Inverse kinematics (IK) is a much more difficult problem than forward
kinematics due to the presence of singularities and nonlinearities. Completely analytical
solutions exist only for a small class of kinematically simple manipulators [
2
]. For many
Algorithms 2022, 15, 469. https://doi.org/10.3390/a15120469 https://www.mdpi.com/journal/algorithms
